# Properties

 Label 16.0.535...928.1 Degree $16$ Signature $[0, 8]$ Discriminant $5.358\times 10^{27}$ Root discriminant $54.08$ Ramified primes $2, 13, 17$ Class number $2448$ (GRH) Class group $[2, 2, 6, 102]$ (GRH) Galois group $C_{16} : C_2$ (as 16T22)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 85*x^14 + 2720*x^12 + 41531*x^10 + 311899*x^8 + 1087983*x^6 + 1861704*x^4 + 1531309*x^2 + 485537)

gp: K = bnfinit(x^16 + 85*x^14 + 2720*x^12 + 41531*x^10 + 311899*x^8 + 1087983*x^6 + 1861704*x^4 + 1531309*x^2 + 485537, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![485537, 0, 1531309, 0, 1861704, 0, 1087983, 0, 311899, 0, 41531, 0, 2720, 0, 85, 0, 1]);

$$x^{16} + 85 x^{14} + 2720 x^{12} + 41531 x^{10} + 311899 x^{8} + 1087983 x^{6} + 1861704 x^{4} + 1531309 x^{2} + 485537$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $16$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[0, 8]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$5357808174640126287142780928$$$$\medspace = 2^{16}\cdot 13^{4}\cdot 17^{15}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $54.08$ sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol))  magma: Abs(Discriminant(Integers(K)))^(1/Degree(K)); Ramified primes: $2, 13, 17$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $8$ This field is not Galois over $\Q$. This is a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{13} a^{10} - \frac{6}{13} a^{8} + \frac{3}{13} a^{6} - \frac{4}{13} a^{4} + \frac{3}{13} a^{2}$, $\frac{1}{13} a^{11} - \frac{6}{13} a^{9} + \frac{3}{13} a^{7} - \frac{4}{13} a^{5} + \frac{3}{13} a^{3}$, $\frac{1}{169} a^{12} - \frac{6}{169} a^{10} + \frac{55}{169} a^{8} + \frac{22}{169} a^{6} - \frac{49}{169} a^{4} + \frac{2}{13} a^{2}$, $\frac{1}{169} a^{13} - \frac{6}{169} a^{11} + \frac{55}{169} a^{9} + \frac{22}{169} a^{7} - \frac{49}{169} a^{5} + \frac{2}{13} a^{3}$, $\frac{1}{35827110046507} a^{14} - \frac{49207453587}{35827110046507} a^{12} - \frac{800979844207}{35827110046507} a^{10} + \frac{8509719033136}{35827110046507} a^{8} - \frac{11485647775495}{35827110046507} a^{6} - \frac{102281126757}{211994734003} a^{4} - \frac{6257481928}{16307287231} a^{2} + \frac{1471088465}{16307287231}$, $\frac{1}{35827110046507} a^{15} - \frac{49207453587}{35827110046507} a^{13} - \frac{800979844207}{35827110046507} a^{11} + \frac{8509719033136}{35827110046507} a^{9} - \frac{11485647775495}{35827110046507} a^{7} - \frac{102281126757}{211994734003} a^{5} - \frac{6257481928}{16307287231} a^{3} + \frac{1471088465}{16307287231} a$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

$C_{2}\times C_{2}\times C_{6}\times C_{102}$, which has order $2448$ (assuming GRH)

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

## Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $7$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-1$$ (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$3640.01221338$$ (assuming GRH) sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{8}\cdot 3640.01221338 \cdot 2448}{2\sqrt{5357808174640126287142780928}}\approx 0.147852824429$ (assuming GRH)

## Galois group

$OD_{32}$ (as 16T22):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 32 The 20 conjugacy class representatives for $C_{16} : C_2$ Character table for $C_{16} : C_2$

## Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

## Sibling fields

 Galois closure: Deg 32

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type R $16$ $16$ $16$ $16$ R R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:

sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:

gp: idealfactors = idealprimedec(K, p); \\ get the data

gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:

magma: idealfactors := Factorization(p*Integers(K)); // get the data

magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];

## Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4} 2.8.8.4x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4} 13.4.2.2x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2} 13.4.0.1x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
17Data not computed